of f. d . semisyik Lie algebra , Max foral subalgebra2-,root R
,
base s --
Ed ... .del,
basis Sea,fakeRt } u shakes}
as fixed last tmi .ei
,hi, fi are Edi , hai , Fa .-
Let cij.
. Hi, dj )
we showed larttni these sentries for it . >e
garate og as aLie algebra . RELATIONS ?
C--i,j= , . . -e
fatal Matt ← Nole I've followed Humphreys ' detention
here . - - modern stuff treespose this matrix! !
Senne's Theorem of is generated by eichi.fi Ci-
- i , - -ie ) subject only to
the relations tisj c-I : ④ [ hi , hj ] -- O
⑤ Lei , f;] -- Sijhi
Serre -0④ [ hi, ej ) - cjiej ,
[hi, fjT= - g- ifj
relations . . . depend Cade)t5i (ej ) -- O
only on Carton matic cadq.j-cjiff.gg = o } for i #J
Note here we're talking about generator and relation for Lee'
algebras .
This requires notion of free tee algebra on a set . See HW 4,05 .
The theorem gives a penetration for aLie algebra of
CC)
g (c) =Flee , hi .fi/i-- Ii
-- ie ) ← free Ge' algebra
-
on these generators
(Sl , 5453, Sif , Sij ) ← ideal generated byeleuuhcanagpoudiigto relate-
re
Serre 's Theorem'
Let C be a Cartan mature armies from a root system
R . The lie algebra GCC) defined by the Senne relations is'
a
f- - d . Semisnie lie algebra with maximal tool subalgebra 2- = Chic -
, he ?
Not system R with sysie rout hi , - , he carpooling to hi.- she , and
Catan matrix G- (@iihTDisj.i. . , e -
Some rule 2 exaples
④ se. e- (% ! ) e .
-
- C! ! !)futz thereposes , h
,-
- diag Ch- to),hedvig 191, -D
Senne relations ⇐ Ee , en) ) = Centered)--O sanitary for f 's
Csis ) Tartar manic A- (ICade Cej ) :O
If in UG) you often seeEee Ee , eid ] -o tuned its
associative algebra relation' e,@ ez-ere ,) - Ce ,ez-ere ,)e ,
-
- O
(112) -21121 ) t 412)
Bz Sos (I ,-
E )@dei5CeD-0eneuhi.hr.f, ,fz (ad EP Ce
, ) = O
←[eater -629333=0 Sene relation when cji=
-2
Expanded : (2221/-312212)+3121223-4222)
Gi (31) @deD4ezt-oCadeg2cen-oTCLeTEEeied3D.o(11114-441121)+641211)-442111) (
211 'D
AA, (12-22) Serre presentation still makes sense n
g,generated Carton matrix SCC) Kae -Moody Slz
"
Proof of Serre's Theorem (assuming Senne 's Theorem 1)
Starting with g ! We've shown gis generated by ee
,hi, fi's
.
Let 's cheek all Sene relationshold a og .
(Sl ) [ hi ,hj )-
-
OV Z is Abehai
(S2) Lei , fi)-- hi V Eee , fj ) -- O
ifjV
--
637 [hi , ej)-
- djfhilej-← Kein Chi - Lj) -wt space
4- which is zero as Li-Lj ER .
( Lj ,di )-
- Cj : £-0as dj-hier
(Sig ) LF -2, Lemma 3 : hi -sting through Lj . Let r,qZ0maxwil so that
dj-r.li , djtqdi are roots. Then *q = Lj Chi )
= cji . q=- g- i
Ligia-
Ta-gin. m%;m? financialmountain.(sis) Similar % Cade icejj -- o ✓
-
Now it follows hat ogCC) → og
ei,hi, fi# eichi.fi
To show this is an csaminphiin , remainsto see devi g
-
- dig
PIRI t IN
This follows from Serre Theorem#-P
Mari thing left to do Ching ! )
Instead of proving that now,let's see the applications .
Serre 's Theorem ' ii really an EXISTENCE THEOREM -every
root system is realized as the root system of some f.d - semisgive og .
Lemma Let R and R' be two root systems with bases
b -
- Sh.. > de } aid D
'-
- { hi , - - -, he'
,} and Catan matrices
( = ( Chi,dj )) aid C'= ( Chi , dj
")) - Assume Rand R'are
NIT noir -phi root systems.Ther og CC) # DCC ') .
Proof WLOG assume R and R'are both hdeiaposable .
Suppose GCC) IgG') .
Then din g= IRI t 1st = IR
't -1-101 -
-
dri ga) .
Moreau,the duration of
smallest inediblenon -Hmat DC) -nodule
is as bated a table n -i ite the same as for ya
') .
It remains to note that there two nisiiaih determine the isomorphic
type of not R unquietf
seet ' R demission of smallest non-trivialiniep.IR/tlDlt#Fl2t2esteak :# Yet
'- Ye:L Taaemw.
Ce follow from the
Debit 21 212- lWeyl diversion
EG 27 78 formula (not
133 poured a- thisEa 56
course ! ) .
F-8
248248
i:L : .l
Conjugacy Theorem Let y be a f -d . seminpli Lie algebra .
All manorial tool subalgebra of 9are Cayuga't mole automorphisms of og .
Proof .
Let 2- and 7'be two max . tonal subalgebra, of og
ol lok,s,C R
'
,I,c
'
ogto ga)
Sene's Theorem showsz→
chic . . he>
Lemma : g Igcc )T gtfo )ti-
ahi ,- iie?So Raid R
'are isomorphic .
Pick uongiui R→ R' taking A→ D
' (ok as all bales are conjugate)
Now →
This obviously advices of⇒ DCC
') taking f
ch, . - , he>
'→ Chi , - . . her ,> . give automorphic og
og taking 2- to 74
Isomorphism Theorem there's mutually Weise bijection's
→ Groot systems}{ f. d. semispite lie algebras} ~ €~←
=
In the forward dueikci , youtake og , pick a
maximal tonal subalgebra 7,
let R be the commanding root system airing fo-Carton decapoatn
of gwet t . This is well -denied independent of Choco q
-2 due
to conjugacy theorem.
In revere duets , take a base D for R ,here
,Carta matric .
Seed 12 to g.CC) , unig Sene 's theme'm .
Etf -- id G aid follow funServe theorem
and Senne theorem '
¢
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